Ensure you have a Twilio account set up and that you have the necessary API credentials. Log in to your Twilio console and navigate to the dashboard to find your Account SID and Auth Token. These will be necessary for accessing Twilio's REST API.

Install the necessary Python libraries to interact with Twilio and MySQL. You can use `pip` to install these libraries:
```bash
pip install twilio mysql-connector-python
```
The `twilio` library will be used for accessing Twilio's REST API, while `mysql-connector-python` will be used for connecting to the MySQL database.

Use the Twilio REST API to fetch the data you need. For example, to retrieve SMS messages, you can use the following Python script:
```python
from twilio.rest import Client
# Your Account SID and Auth Token from twilio.com/console
account_sid = 'your_account_sid'
auth_token = 'your_auth_token'
client = Client(account_sid, auth_token)
# Fetch messages
messages = client.messages.list()
for message in messages:
print(message.sid, message.body)
```
Modify the script to extract other types of data as needed.

Ensure you have a MySQL database set up and ready to receive data. Create a table with the appropriate schema to store the data from Twilio. For example, to store SMS message data, your SQL table creation statement might look like:
```sql
CREATE TABLE messages (
id INT AUTO_INCREMENT PRIMARY KEY,
sid VARCHAR(50),
body TEXT,
date_sent DATETIME
);
```

Use Python to connect to your MySQL database. Here's an example script for connecting:
```python
import mysql.connector
connection = mysql.connector.connect(
host='your_host',
user='your_username',
password='your_password',
database='your_database'
)
cursor = connection.cursor()
```

With the data retrieved from Twilio and your database connection established, insert the data into your MySQL table. Here's an example continuation of your Python script:
```python
for message in messages:
cursor.execute(
"INSERT INTO messages (sid, body, date_sent) VALUES (%s, %s, %s)",
(message.sid, message.body, message.date_sent)
)
connection.commit()
```

After inserting the data, verify that the transfer was successful by querying the MySQL database. You can use a simple SELECT statement to confirm the data:
```python
cursor.execute("SELECT * FROM messages")
for (sid, body, date_sent) in cursor.fetchall():
print(sid, body, date_sent)
```
Ensure that all expected data from Twilio is now present in your MySQL table.
By following these steps, you should be able to effectively move data from Twilio to a MySQL database using Python, without relying on third-party connectors or integrations.